Probability of range of values

2 In a certain large college, \(22 \%\) of students own a car.

1. 3 students from the college are chosen at random. Find the probability that all 3 students own a car.
2. 16 students from the college are chosen at random. Find the probability that the number of these students who own a car is at least 2 and at most 4 .

1 Biscuits are sold in packets of 18. There is a constant probability that any biscuit is broken, independently of other biscuits. The mean number of broken biscuits in a packet has been found to be 2.7 . Find the probability that a packet contains between 2 and 4 (inclusive) broken biscuits.

1 A fair die is thrown 10 times. Find the probability that the number of sixes obtained is between 3 and 5 inclusive. 2120 people were asked to read an article in a newspaper. The times taken, to the nearest second, by the people to read the article are summarised in the following table. Calculate estimates of the mean and standard deviation of the reading times.

5

1. \(20 \%\) of people in the large town of Carnley support the Residents' Party. 12 people from Carnley are selected at random. Out of these 12 people, the number who support the Residents' Party is denoted by \(U\). Find
   (a) \(\mathrm { P } ( U \leqslant 5 )\),
   (b) \(\quad \mathrm { P } ( U \geqslant 3 )\).
2. \(30 \%\) of people in Carnley support the Commerce Party. 15 people from Carnley are selected at random. Out of these 15 people, the number who support the Commerce Party is denoted by \(V\). Find \(\mathrm { P } ( V = 4 )\).

3 In a supermarket the proportion of shoppers who buy washing powder is denoted by \(p .16\) shoppers are selected at random.

1. Given that \(p = 0.35\), use tables to find the probability that the number of shoppers who buy washing powder is
   (a) at least 8,
   (b) between 4 and 9 inclusive.
2. Given instead that \(p = 0.38\), find the probability that the number of shoppers who buy washing powder is exactly 6 .

6 Plastic clothes pegs are made in various colours.
The number of red pegs may be modelled by a binomial distribution with parameter \(p\) equal to 0.2 . The contents of packets of 50 pegs of mixed colours may be considered to be random samples.

1. Determine the probability that a packet contains:
   1. less than or equal to 15 red pegs;
   2. exactly 10 red pegs;
   3. more than 5 but fewer than 15 red pegs.
2. Sly, a student, claims to have counted the number of red pegs in each of 100 packets of 50 pegs. From his results the following values are calculated. Mean number of red pegs per packet \(= 10.5\)
   Variance of number of red pegs per packet \(= 20.41\)
   Comment on the validity of Sly's claim.

4 Clay pigeon shooting is the sport of shooting at special flying clay targets with a shotgun.

1. Rhys, a novice, uses a single-barrelled shotgun. The probability that he hits a target is 0.45 , and may be assumed to be independent from target to target. Determine the probability that, in a series of shots at 15 targets, he hits:
   1. at most 5 targets;
   2. more than 10 targets;
   3. exactly 6 targets;
   4. at least 5 but at most 10 targets.
2. Sasha, an expert, uses a double-barrelled shotgun. She shoots at each target with the gun's first barrel and, only if she misses, does she then shoot at the target with the gun's second barrel. The probability that she hits a target with a shot using her gun's first barrel is 0.85 . The conditional probability that she hits a target with a shot using her gun's second barrel, given that she has missed the target with a shot using her gun's first barrel, is 0.80 . Assume that Sasha's shooting is independent from target to target.
   1. Show that the probability that Sasha hits a target is 0.97 .
   2. Determine the probability that, in a series of shots at 50 targets, Sasha hits at least 48 targets.
   3. In a series of shots at 80 targets, calculate the mean number of times that Sasha shoots at targets with her gun's second barrel.
      \includegraphics[max width=\textwidth, alt={}]{156f9453-ebc6-4406-b5bc-08d1918ebc62-14_2486_1714_221_153}

4 The records at a passport office show that, on average, 15 per cent of photographs that accompany applications for passport renewals are unusable. Assume that exactly one photograph accompanies each application.

1. Determine the probability that in a random sample of 40 applications:
   1. exactly 6 photographs are unusable;
   2. at most 5 photographs are unusable;
   3. more than 5 but fewer than 10 photographs are unusable.
2. Calculate the mean and the standard deviation for the number of photographs that are unusable in a random sample of \(\mathbf { 3 2 }\) applications.
3. Mr Stickler processes 32 applications each day. His records for the previous 10 days show that the numbers of photographs that he deemed unusable were $$\begin{array} { l l l l l l l l l l } 8 & 6 & 10 & 7 & 9 & 7 & 8 & 9 & 6 & 7 \end{array}$$ By calculating the mean and the standard deviation of these values, comment, with reasons, on the suitability of the \(\mathrm { B } ( 32,0.15 )\) model for the number of photographs deemed unusable each day by Mr Stickler.

3 Stopoff owns a chain of hotels. Guests are presented with the bills for their stays when they check out.

1. Assume that the number of bills that contain errors may be modelled by a binomial distribution with parameters \(n\) and \(p\), where \(p = 0.30\). Determine the probability that, in a random sample of 40 bills:
   1. at most 10 bills contain errors;
   2. at least 15 bills contain errors;
   3. exactly 12 bills contain errors.
2. Calculate the mean and the variance for each of the distributions \(\mathrm { B } ( 16,0.20 )\) and \(B ( 16,0.125 )\).
3. Stan, who is a travelling salesperson, always uses Stopoff hotels. He holds one of its diamond customer cards and so should qualify for special customer care. However, he regularly finds errors in his bills when he checks out. Each month, during a 12-month period, Stan stayed in Stopoff hotels on exactly 16 occasions. He recorded, each month, the number of occasions on which his bill contained errors. His recorded values were as follows. $$\begin{array} { l l l l l l l l l l l l } 2 & 1 & 4 & 3 & 1 & 3 & 0 & 3 & 1 & 0 & 5 & 1 \end{array}$$
   1. Calculate the mean and the variance of these 12 values.
   2. Hence state with reasons which, if either, of the distributions \(\mathrm { B } ( 16,0.20 )\) and \(B ( 16,0.125 )\) is likely to provide a satisfactory model for these 12 values.

6 Each weekday, Monday to Friday, Trina catches a train from her local station. She claims that the probability that the train arrives on time at the station is 0.4 , and that the train's arrival time is independent from day to day.

1. Assuming her claims to be true, determine the probability that the train arrives on time at the station:
   1. on at most 3 days during a 2 -week period ( 10 days);
   2. on more than 10 days but fewer than 20 days during an 8-week period.
2. 1. Assuming Trina's claims to be true, determine the mean and standard deviation for the number of times during a week (5 days) that the train arrives on time at the station.
   2. Each week, for a period of 13 weeks, Trina's travelling colleague, Suzie, records the number of times that the train arrives on time at the station. Suzie's results are

      2

      2

      4

      1

      2

      3

      3

      2

      2

      0

      3

      2

      0

      Calculate the mean and standard deviation of these values.
   3. Hence comment on the likely validity of Trina's claims.

6 For the adult population of the UK, 35 per cent of men and 29 per cent of women do not wear glasses or contact lenses.

1. Determine the probability that, in a random sample of 40 men:
   1. at most 15 do not wear glasses or contact lenses;
   2. more than 10 but fewer than 20 do not wear glasses or contact lenses.
2. Calculate the probability that, in a random sample of 10 women, exactly 3 do not wear glasses or contact lenses.
3. 1. Calculate the mean and the variance for the number who do wear glasses or contact lenses in a random sample of 20 women.
   2. The numbers wearing glasses or contact lenses in 10 groups, each of 20 women, had a mean of 16.5 and a variance of 2.50. Comment on the claim that these 10 groups were not random samples.

7 Mr Alott and Miss Fewer work in a postal sorting office.

1. The number of letters per batch, \(R\), sorted incorrectly by Mr Alott when sorting batches of 50 letters may be modelled by the distribution \(\mathrm { B } ( 50,0.15 )\). Determine:
   1. \(\mathrm { P } ( R < 10 )\);
   2. \(\mathrm { P } ( 5 \leqslant R \leqslant 10 )\).
2. It is assumed that the probability that Miss Fewer sorts a letter incorrectly is 0.06 , and that her sorting of a letter incorrectly is independent from letter to letter.
   1. Calculate the probability that, when sorting a batch of \(\mathbf { 2 2 }\) letters, Miss Fewer sorts exactly 2 letters incorrectly.
   2. Calculate the probability that, when sorting a batch of \(\mathbf { 3 5 }\) letters, Miss Fewer sorts at least 1 letter incorrectly.
   3. Calculate the mean and the variance for the number of letters sorted correctly by Miss Fewer when she sorts a batch of \(\mathbf { 1 2 0 }\) letters.
   4. Miss Fewer sorts a random sample of 20 batches, each containing 120 letters. The number of letters sorted correctly per batch has a mean of 112.8 and a variance of 56.86 . Comment on the assumptions that the probability that Miss Fewer sorts a letter incorrectly is 0.06 , and that her sorting of a letter incorrectly is independent from letter to letter.
      \includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-15_2484_1709_223_153}

4 In a certain country, 15 per cent of the male population is left-handed.

1. Determine the probability that, in a random sample of 50 males from this country:
   1. at most 10 are left-handed;
   2. at least 5 are left-handed;
   3. more than 6 but fewer than 12 are left-handed.
2. In the same country, 11 per cent of the female population is left-handed. Calculate the probability that, in a random sample of 35 females from this country, exactly 4 are left-handed.
3. A sample of 2000 people is selected at random from the population of the country. The proportion of males in the sample is 52 per cent. How many people in the sample would you expect to be left-handed?
   \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-09_2484_1709_223_153}

6 An amateur tennis club purchases tennis balls that have been used previously in professional tournaments. The probability that each such ball fails a standard bounce test is 0.15 . The club purchases boxes each containing 10 of these tennis balls. Assume that the 10 balls in any box represent a random sample.

1. Determine the probability that the number of balls in a box which fail the bounce test is:
   1. at most 2 ;
   2. at least 2;
   3. more than 1 but fewer than 5 .
2. Determine the probability that, in \(\mathbf { 5 }\) boxes, the total number of balls which fail the bounce test is:
   1. more than 5 ;
   2. at least 5 but at most 10 .

5 Each evening Aaron sets his alarm for 7 am. He believes that the probability that he wakes before his alarm rings each morning is 0.4 , and is independent from morning to morning.

1. Assuming that Aaron's belief is correct, determine the probability that, during a week (7 mornings), he wakes before his alarm rings:
   1. on 2 or fewer mornings;
   2. on more than 1 but fewer than 5 mornings.
2. Assuming that Aaron's belief is correct, calculate the probability that, during a 4 -week period, he wakes before his alarm rings on exactly 7 mornings.
3. Assuming that Aaron's belief is correct, calculate values for the mean and standard deviation of the number of mornings in a week when Aaron wakes before his alarm rings.
   (2 marks)
4. During a 50-week period, Aaron records, each week, the number of mornings on which he wakes before his alarm rings. The results are as follows.

   Number of mornings	0	1	2	3	4	5	6	7
   Frequency	10	8	7	7	5	5	4	4

   1. Calculate the mean and standard deviation of these data.
   2. State, giving reasons, whether your answers to part (d)(i) support Aaron's belief that the probability that he wakes before his alarm rings each morning is 0.4 , and is independent from morning to morning.
      (3 marks)

6 During the winter, the probability that Barry's cat, Sylvester, chooses to stay outside all night is 0.35 , and the cat's choice is independent from night to night.

1. Determine the probability that, during a period of 2 weeks ( 14 nights) in winter, Sylvester chooses to stay outside:
   1. on at most 7 nights;
   2. on at least 11 nights;
   3. on more than 5 nights but fewer than 10 nights.
2. Calculate the probability that, during a period of \(\mathbf { 3 }\) weeks in winter, Sylvester chooses to stay outside on exactly 4 nights.
3. Barry claims that, during the summer, the number of nights per week, \(S\), on which Sylvester chooses to stay outside can be modelled by a binomial distribution with \(n = 7\) and \(p = \frac { 5 } { 7 }\).
   1. Assuming that Barry's claim is correct, find the mean and the variance of \(S\).
   2. For a period of 13 weeks during the summer, the number of nights per week on which Sylvester chose to stay outside had a mean of 5 and a variance of 1.5 . Comment on Barry's claim.
      (2 marks)

5 Kirk and Les regularly play each other at darts.

1. The probability that Kirk wins any game is 0.3 , and the outcome of each game is independent of the outcome of every other game. Find the probability that, in a match of 15 games, Kirk wins:
   1. exactly 5 games;
   2. fewer than half of the games;
   3. more than 2 but fewer than 7 games.
2. Kirk attends darts coaching sessions for three months. He then claims that he has a probability of 0.4 of winning any game, and that the outcome of each game is independent of the outcome of every other game.
   1. Assuming this claim to be true, calculate the mean and standard deviation for the number of games won by Kirk in a match of 15 games.
   2. To assess Kirk's claim, Les keeps a record of the number of games won by Kirk in a series of 10 matches, each of 15 games, with the following results: $$\begin{array} { l l l l l l l l l l } 8 & 5 & 6 & 3 & 9 & 12 & 4 & 2 & 6 & 5 \end{array}$$ Calculate the mean and standard deviation of these values.
   3. Hence comment on the validity of Kirk's claim.

6

1. In a particular country, 35 per cent of the population is estimated to have at least one mobile phone. A sample of 40 people is selected from the population.
   Use the distribution \(\mathrm { B } ( 40,0.35 )\) to estimate the probability that the number of people in the sample that have at least one mobile phone is:
   1. at most 15 ;
   2. more than 10 ;
   3. more than 12 but fewer than 18 ;
   4. exactly equal to the mean of the distribution.
2. In the same country, 70 per cent of households have a landline telephone connection. A sample of 50 households is selected from all households in the country.
   Stating a necessary condition regarding this selection, estimate the probability that fewer than 30 households have a landline telephone connection.
   [0pt] [4 marks]

3 In a supermarket the proportion of shoppers who buy washing powder is denoted by \(p\). 16 shoppers are selected at random.

1. Given that \(p = 0.35\), use tables to find the probability that the number of shoppers who buy washing powder is
   (a) at least 8,
   (b) between 4 and 9 inclusive.
2. Given instead that \(p = 0.38\), find the probability that the number of shoppers who buy washing powder is exactly 6 . \section*{June 2005}

16
A mathematical puzzle is published every day in a newspaper.
Over a long period of time Paula is able to solve the puzzle correctly \(60 \%\) of the time.
16

1. For a randomly chosen 14-day period find the probability that:
   16
2. 1. 
      Paula correctly solves exactly 8 puzzles
      [0pt] [1 mark]
      16
3. (ii)
   Paula correctly solves at least 7 but not more than 11 puzzles.
   [0pt] [2 marks]
   16
4. 
   \end{tabular}
   \hline \end{tabular} \end{center}

15 It is given that $$X \sim \mathrm {~B} ( 48,0.175 )$$ 15

1. Find the mean of \(X\)
   [0pt] [1 mark] 15
2. Show that the variance of \(X\) is 6.93
   [0pt] [1 mark] 15
3. Find \(\mathrm { P } ( X < 10 )\)
   [0pt] [1 mark] 15
4. \(\quad\) Find \(\mathrm { P } ( X \geq 6 )\)
   [0pt] [2 marks]
   15
5. \(\quad\) Find \(\mathrm { P } ( 9 \leq X \leq 15 )\)
   [0pt] [2 marks] L
   15
6. The aeroplanes used on a particular route have 48 seats. The proportion of passengers who use this route to travel for business is known to be 17.5\% Make two comments on whether it would be appropriate to use \(X\) to model the number of passengers on an aeroplane who are travelling for business using this route.